ANNUAIRE DE L’UNIVERSITE DE SOFIA “St. Kl. OHRIDSKI”

FACULTE DE MATHEMATIQUES ET INFORMATIQUE

NORMALLY GENERATED SUBSPACES OF LOGARITHMICCANONICAL SECTIONS

BORIS KOTZEV, AZNIV KASPARIAN 1

The logarithmic-canonical bundle Ω2A′ (T

′) of a smooth toroidal compactification A′ =(B/Γ)′ of a ball quotient B/Γ is known to be sufficiently ample over the Baily-Borel

compactification A = B/Γ. The present work develops criteria for a subspace V ⊆H0(A′,Ω2

A′ (T′)) to be normally generated over A, i.e., to determine a regular immer-

sive projective morphism of A with normal image. These are applied to a specificexample A′

1 = (B/Γ1)′ over the Gauss numbers. The first section organizes some pre-

liminaries. The second one provides two sufficient conditions for the normal generationof a subspace V ⊆ H0(A′,Ω2

A′ (T′)).

Keywords. Modular forms, holomorphic sections of line bundles, abelian functions,sufficiently ample and normally generated line bundles.

2000 Math. Subject Classification. main 11F23, secondary 14G35

1. PRELIMINARIES

Throughout, let B = z = (z1, z2) ∈ C2 | |z1|2 + |z2|2 < 1 = SU2,1/S(U2 × U1)be the complex two dimensional ball and Γ ⊂ SU2,1 be a lattice, acting freely on B.The compact B/Γ are of general type. The non-compact B/Γ admit smooth toroidalcompactification (B/Γ)′ by a disjoint union T ′ = ∪h

i=1T′i of smooth irreducible

elliptic curves T ′i . From now on, we concentrate on A′ = (B/Γ)′ with abelian

minimal model A. In such a case, the lattice Γ, the ball quotient B/Γ and itscompactifications are said to be co-abelian.

The contraction ξ : A′ → A of the rational (−1)-curves on A′ restricts to abiregular morphism ξ : T ′

i → ξ(T ′i ) = Ti, as far as an abelian surface A does

not support rational curves. In such a way, ξ produces the multi-elliptic divisor

1Research partially supported by Contract 171 / 05. 2008

T = ξ(T ′) =h∑

i=1

Ti ⊂ A, i.e., a divisor with smooth elliptic irreducible components

Ti. According to Kobayashi hyperbolicity of B/Γ, any irreducible component of theexceptional divisor of ξ intersects T ′ in at least two points. Therefore ξ : A′ → A isthe blow-up of A at the singular locus T sing =

∑1≤i<j≤h

Ti ∩ Tj of T . Holzapfel has

shown in [5] that the blow-up A′ of an abelian surface A at the singular locus T sing =∑1≤i<j≤h

Ti∩Tj of a multi-elliptic divisor T =h∑

i=1

Ti is the toroidal compactification

A′ = (B/Γ)′ of a smooth ball quotient B/Γ if and only if A = E×E is the Cartesiansquare of an elliptic curve E and

h∑i=1

card(Ti ∩ T sing) = 4card(T sing). (1.1)

In order to describe the smooth irreducible elliptic curves Ti on A and their in-tersections, let us note that the inclusions Ti ⊂ A = E × E are morphisms ofabelian varieties. Consequently, they lift to affine linear maps of the correspondinguniversal covers and

Ti = (u+ π1(E), v + π1(E)) | aiu+ biv + ci ∈ π1(E) =

for some ai, bi, ci ∈ C. The fundamental group

π1(Ti) = t ∈ C | bit+ π1(E) = −ait+ π1(E) = π1(E) = a−1i π1(E) ∩ b−1

i π1(E).

If Γ is an arithmetic lattice then the elliptic curve E has complex multiplicationby an imaginary quadratic number field K = Q(

√−d), d ∈ N. As a result, Γ is

commensurable with the full Picard modular group SU2,1(O−d) over the integersring O−d of Q(

√−d). Such Γ are called Picard modular groups. Moreover, all

Ti are defined over K. For simplicity, we assume that π1(E) = O−d, in orderto have maximal endomorphism ring End(E) = O−d. Since K = Q(

√−d) is

the fraction field of O−d, one can choose ai, bi ∈ O−d. Thus, π1(Ti) ⊇ O−d,aiπ1(E) + biπ1(E) ⊆ O−d and Ti has minimal fundamental group π1(Ti) = O−d

exactly when aiπ1(E) + biπ1(E) = π1(E) = O−d. In particular, if K is of classnumber 1, then all the smooth elliptic curves Ti ⊂ A = C2/ (O−d ×O−d), definedover K = Q(

√−d), have minimal fundamental groups π1(Ti) = O−d. From now

on, we do not restrict the class number of K = Q(√−d), but confine only to smooth

irreducible elliptic curves Ti with minimal fundamental groups π1(Ti) = π1(E) =O−d. If bi = 0 then

T(1)i = (bit+ π1(E),−ait− b−1

i ci + π1(E)) | t ∈ C ⊆ Ti.

Moreover, the complete pre-image of T(1)i in the universal cover A = C2 of A is

π1(Ti)-invariant family of complex lines. Therefore T(1)i is an elliptic curve and

coincides with Ti.The notations from the next lemma will be used throughout:

2

Lemma 1. Let Ts = (u + O−d, v + O−d) | asu + bsv + cs ∈ Od and Ds =(u+O−d, v+O−d) | asu+bsv+cs+µs ∈ O−d for 1 ≤ s ≤ 3 be elliptic curves withminimal fundamental groups π1(Ts) = π1(Ds) = O−d on A = (C/O−d)× (C/O−d)and

∆ij := det

(ai ajbi bj

), ∆ := det

a1 a2 a3b1 b2 b3c1 c2 c3

.

Then for any even permutation i, j, l of 1, 2, 3 there hold the following:

(i) the intersection number Ti.Tj = NQ(

√−d)

Q (∆ij), where NQ(

√−d)

Q : Q(√−d) →

Q stands for the norm;(ii) Ti ∩ Tj ⊂ Dl if and only if µl ∈ O−d − ∆−1

ij ∆ and ∆−1ij ∆jl,∆

−1ij ∆li ∈

End(E) = O−d;(iii) T1 ∩ T2 ∩ T3 = ∅ if and only if ∆ ∈ ∆12O−d +∆23O−d +∆31O−d.

Proof. (i) If Ti ∩ Tj = ∅ then the liftings of Ti, Tj to the universal cover A = C2

of A are discrete families of mutually parallel lines. In such a case, we say brieflythat Ti and Tj are parallel. That allows to choose aj = ai, bj = bi and to calculate

that NQ(

√−d)

Q (∆ij) = NQ(

√−d)

Q (0) = 0 = Ti.Tj . When Ti ∩ Tj = ∅, one can movethe origin oA = (oE , oE) ∈ A in Ti ∩ Tj and represent

Ti = (bit+O−d,−ait+O−d) | t ∈ C, Tj = (u+O−d, v+O−d) | aju+bjv ∈ O−d.

Then the intersection

Ti ∩ Tj = (bit+O−d,−ait+O−d) |∆ijt ∈ O−d ⊂ C ≃

(∆−1ij O−d)/(b

−1i O−d ∩ a−1

i O−d) =(∆−1

ij O−d

)/O−d ≃ O−d/∆ijO−d.

For an arbitrary lattice Λ ⊂ C, let us denote by F(Λ) a Λ-fundamental domain onC. As far as F(∆ijO−d) is the O−d/∆ijO−d-orbit of F(O−d), the index

[O−d : ∆ijO−d] =volF(∆ijO−d)

volF(O−d)=

volF(|∆ij |O−d)

volF(O−d)= |∆ij |2 = N

Q(√−d)

Q (∆ij).

(ii) The intersection Ti ∩ Tj consists of the π1(A)-equivalence classes of thesolutions (u, v) ∈ C2 of ∣∣∣∣ aiu+ biv = λ1 − ci

aju+ bjv = λ2 − cj

for arbitrary λ1, λ2 ∈ π1(E) = O−d. A point

(∆−1ij (bicj − bjci) + ∆−1

ij (bjλ1 − biλ2), ∆−1ij (ajci − aicj) + ∆−1

ij (aiλ2 − ajλ1))

belongs to the lifting ofDl iff−∆−1ij ∆jlλ1−∆−1

ij ∆liλ2+∆−1ij (ci∆jl+cj∆li)+cl+µl =

−∆−1ij ∆jlλ1−∆−1

ij ∆liλ2+∆−1ij ∆+µl ∈ π1(E) = O−d for ∀λ1, λ2 ∈ π1(E). That, in

turn, is equivalent to ∆−1ij ∆+µl ∈ π1(E) = O−d and ∆−1

ij ∆jl,∆−1ij ∆li ∈ End(E) =

O−d.

3

(iii) For arbitrary λ1, λ2, λ3 ∈ π1(E) = O−d, the linear system∣∣∣∣∣∣a1u+ b1v = λ1 − c1a2u+ b2v = λ2 − c2a3u+ b3v = λ3 − c3

has no solutions exactly when

det

a1 b1 λ1 − c1a2 b2 λ2 − c2a3 b3 λ3 − c3

= ∆23λ1 +∆31λ2 +∆12λ3 −∆ = 0.

The non-arithmetic lattices Γ ⊂ SU2,1 correspond to abelian surfaces A = E×E,whose elliptic factors E have minimal endomorphism rings End(E) = Z. Then theliftings of the elliptic curves Ti ⊂ A with π1(Ti) = π1(E) to the universal coverA = C2 of A are given by aiu + biv + ci ∈ π1(E) with ai, bi ∈ Z. As a result, the

intersection numbers Ti.Tj = NQ(

√−d)

Q (∆ij) are comparatively large and there are

very few chances for construction of a multi-elliptic divisor T =h∑

i=1

⊂ A, subject to

(1.1). This is a sort of a motivation for restricting our attention to the arithmeticcase.

The smooth irreducible elliptic curves T ′i ⊂ A′ contract to the Γ-orbits κi =

Γ(p) ∈ ∂ΓB/Γ of the Γ-rational boundary points p ∈ ∂ΓB. These κi are called

cusps. The resulting Baily-Borel compactification A = B/Γ = (B/Γ) ∪ (∂ΓB/Γ) isa normal projective surface.

Definition 2. Let Γ be a Picard modular group, γ ∈ Γ and Jac(γ) = ∂(γ1,γ2)∂(z1,z2)

be the Jacobian matrix of γ = (γ1, γ2) : B → B ⊂ C2. The global holomorphicfunctions δ : B → C with transformation law

γ∗(δ)(z) = δγ(z) = [det Jac(γ)]−n

δ(z) for ∀γ ∈ Γ, ∀z ∈ B

are called Γ-modular forms of weight n.The Γ-modular forms of weight n constitute a C-linear space, which is denoted

by [Γ, n].

Definition 3. A Γ-modular form δ ∈ [Γ, n] is cuspidal if δ(κi) = 0 at all the cuspsκi ∈ ∂ΓB/Γ.

The cuspidal Γ-modular forms of weight n form the subspace [Γ, n]cusp of [Γ, n].

For any natural number n there is a C-linear embedding

jn : H0(B,OB) −→ H0(B,(Ω2

B)⊗n

)jn(δ)(z) = δ(z)(dz1 ∧ dz2)

⊗n

4

of the global holomorphic functions on the ball in the global holomorphic sections

of the n-th pluri-canonical bundle(Ω2

B)⊗n

. It restricts to an isomorphism

jn : [Γ, n] −→ H0(B,(Ω2

B)⊗n

)Γof the Γ-modular forms of weight n with the Γ-invariant holomorphic sections of(Ω2

B)⊗n

. Note that the subspace H0(B,(Ω2

B)⊗n

)Γof H0

(B/Γ,

(Ω2

B/Γ

)⊗n)

acts

on A = B/Γ, extending over the cusps ∂ΓB/Γ of codimension 2 in A.The tensor product Ω2

A′(T ′) = Ω2A′ ⊗C OA′(T ′) is called logarithmic canonical

bundle of A′, while Ω2A′(T ′)⊗n are referred to as logarithmic pluri-canonical bundles.

Hemperly has observed in [3] that

jn[Γ, n] = H0(B,(Ω2

B)⊗n

)Γ= H0

(A′,Ω2

A′(T ′)⊗n),

as long as the holomorphic sections from these spaces have one and a same coordi-nate transformation law. A classical result of Baily-Borel establishes that Ω2

A′(T ′)

is sufficiently ample on A. The present article provides sufficient conditions for theampleness of Ω2

A′(T ′) on A.Note that the canonical bundle KA′ = ξ∗KA + OA′(L) = ξ∗OA + OA′(L) =

OA′(L) is associated with the exceptional divisor L = ξ−1(T sing) of ξ : A′ → A. Ifs is a global meromorphic section of Ω2

A′ and t is a global meromorphic section ofOA′(T ′) then the tensoring

(s⊗C t)⊗(−n)

: H0(A′,Ω2

A′(T ′)⊗n)−→ LA′(n(L+ T ′))

is a C-linear isomorphism with

LA′(n(L+ T ′)) = f ∈ Mer(A′) | (f) + n(L+ T ′) ≥ 0.

The isomorphism ξ∗ : Mer(A) → Mer(A′) of the meromorphic function fields in-duces a linear isomorphism

(ξ∗)−1

: LA′(n(L+ T ′)) −→ LA(nT, nTsing),

where mp : Div(A) → Z stands for the multiplicity at a point p ∈ A and

LA(nT, nTsing) = f ∈ Mer(A) | (f) + nT ≥ 0, mp(f) + n ≥ 0 for ∀p ∈ T sing.

The linear isomorphisms

τn := (ξ∗)−1(s⊗C t)⊗(−n) : jn[Γ, n] −→ LA(nT, nTsing)

are called transfers of modular forms of weight n to abelian functions.

5

For any δ ∈ [Γ, 1], note that δ(κi) = 0 if and only if Ti ⊂ (τ1j1(δ))∞. Observealso that τ1j1[Γ, 1]cusp = f ∈ LA(T, T

sing) | (f)∞ = ∅ = C and fix the cuspidalform ηo = (τ1j1)

−1(1) of weight 1.Towards the construction of abelian functions f ∈ LA(T, T

sing), let us recallfrom [7] that any elliptic function g : E → P1 can be represented as

g(z) = Co

k∏i=1

σ(z − αi)

σ(z − βi), (1.2)

where

σ(z) = z∏

λ∈π1(E)\0

(1− z

λ

) zλ+ 1

2 (zλ )

2

is Weierstrass’ σ-function, αi, βi, Co ∈ C andk∑

i=1

αi ≡k∑

i=1

βi(modπ1(E)). The

points of E = C/π1(E) are of the form a = a + π1(E) for some a ∈ C. Theelliptic function (1.2) takes all the values from P1 with one and a same multiplicityk. Moreover, if g−1(x) = pi(x) ∈ E | 1 ≤ i ≤ k for some x ∈ C ⊂ P1, thenk∑

i=1

pi(x) =k∑

i=1

βi. Observe that σ : C → C is a non-periodic entire function, but its

divisor (σ)C = π1(E) on C is π1(E)-invariant. That enables to define the divisor(σ)E = oE of σ on E. In global holomorphic coordinates (u, v) ∈ C2, the divisor

(σ(aiu+ biv + ci))C2 = (u, v) ∈ C2 | aiu+ biv + ci ∈ π1(E) = O−d

is the complete pre-image of Ti in the universal cover A = C2 of A. That allows todefine the divisor

(σ(aiu+ biv + ci)) = (σ(aiu+ biv + ci))A = Ti.

Let f ∈ LA(T ) be an abelian function with pole divisor (f)∞ =k∑

i=1

Ti, after an

eventual permutation of the irreducible components of T . Then

f∞ :=k∏

i=1

σ(aiu+ biv + ci) and f0 := ff∞ (1.3)

are (non-periodic) entire functions on C2. Let ζ = σ′

σ be Weierstrass’ ζ-function,η : π1(E) → C be the Z-linear homomorphism, satisfying ζ(z + λ) = ζ(z) + η(λ)for all z ∈ C, λ ∈ π1(E) and

ε(λ) =

1 for λ ∈ 2π1(E),

−1 for λ ∈ π1(E) \ 2π1(E).

Recall from [6] the π1(E)-transformation law

σ(z + λ)

σ(z)= ε(λ)eη(λ)(z+

λ2 ) for ∀λ ∈ π1(E), ∀z ∈ C.

6

Under the assumption (1.3), the π1(A)-periodicity of f is equivalent to

f0(u+ λ, v)

f0(u, v)=

f∞(u+ λ, v)

f∞(u, v)=

k∏i=1

ε(aiλ)eη(aiλ)

(aiu+biv+ci+

aiλ

2

)and

f0(u, v + λ)

f0(u, v)=

f∞(u, v + λ)

f∞(u, v)=

k∏i=1

ε(biλ)eη(biλ)

(aiu+biv+ci+

biλ

2

)

for ∀λ ∈ π1(E) = O−d, ∀(u, v) ∈ C2. We choose

f0(u, v) =

k∏i=1

σ(aiu+ biv + ci + µi)

and reduce the π1(A)-periodicity of f to

1 =f(u+ λ, v)

f(u, v)= e

k∑i=1

η(aiλ)µi

, 1 =f(u, v + λ)

f(u, v)= e

k∑i=1

η(biλ)µi

for ∀λ ∈ O−d, ∀(u, v) ∈ C2. Let us mention that Holzapfel has studied f ∈ LA(T )of the above form with at most three non-parallel irreducible components of (f)∞,intersecting pairwise in single points. The next lemma provides a sufficient (but notnecessary) condition for π1(A)-periodicity of a σ-quotient, whose pole divisor hasan arbitrary number of irreducible components with arbitrary intersection numbers.

Lemma 4. Ifk∑

i=1

aiµi =

k∑i=1

aiµi =

k∑i=1

biµi =

k∑i=1

biµi = 0 (1.4)

then the σ-quotient

f(u, v) =

k∏i=1

σ(aiu+ biv + ci + µi)

σ(aiu+ biv + ci)(1.5)

is O−d ×O−d-periodic.

Proof. Let us recall from [1] that the integers ring of an imaginary quadratic numberfield Q(

√−d) is of the form O−d = Z+ 2ωZ for

2ω =

√−d for −d ≡ 1(mod 4),

−1+√−d

2 for −d ≡ 1(mod 4).

Any ν ∈ O−d has unique representation ν = x+ 2ωy with

x =2ων − 2ων

2ω − 2ω∈ Z, y =

ν − ν

2ω − 2ω∈ Z.

7

Legendre’s equalityη(2ω)− 2ωη(1) = 2π

√−1,

(cf.[6]) implies that

η(ν) = νη(1) +ν − ν

2ω − 2ω2π

√−1 for ∀ν ∈ O−d.

As a result,

k∑i=1

η(aiλ)µi =

(k∑

i=1

aiµi

)λη(1) +

(k∑

i=1

aiµi

)2π

√−1λ

2ω − 2ω−

(k∑

i=1

aiµi

)2π

√−1λ

2ω − 2ω.

Mutually parallel smooth elliptic curves T1, . . . , Tk admit liftings

Ti = (u+O−d, v +O−d) | a1u+ b1v + ci ∈ O−d.

For arbitrary µj ∈ C withk∑

i=1

µi = 0, the σ-quotient

f(u, v) =

k∏i=1

σ(a1u+ b1v + ci + µi)

σ(a1u+ b1v + ci)(1.6)

belongs to LA(T, Tsing) and has smooth pole divisor (f)∞ =

k∑i=1

Ti. After [4], we

say that (1.6) is a k-fold parallel σ-quotient. A σ-quotient (1.5) has smooth poledivisor if and only if it is k-fold parallel.

Definition 5. A special σ-quotient of order k is a function of the form (1.5), whichis subject to (1.4), has singular pole divisor (f)∞ and µi ∈ O−d for all 1 ≤ i ≤ k.

Lemma 6. If f ∈ LA(T, Tsing) is a special σ-quotient of order k ≥ 2, then at any

point p ∈ (f)sing∞ the multiplicity

2 ≤ mp(f)∞ ≤[k + 1

2

],

where[k+12

]is the greatest natural number, non-exceeding k+1

2 .In particular, LA(T, T

isng) does not contain a special σ-quotient of order 2.

Proof. The smoothness of the irreducible components T1, . . . , Tk of (f)∞ resultsin (f)sing∞ ⊂

∑1≤i<j≤k

(Ti ∩ Tj) and implies that mp(f)∞ ≥ 2 for all p ∈ (f)sing∞ .

Suppose that mp(f)∞ = m for some 2 ≤ m ≤ k. After an eventual permutation ofT1, . . . , Tk, one can assume that p ∈ T1 ∩ . . . ∩ Tm and p ∈ Tm+1 + . . .+ Tk. Then

mp(f) + 1 = mp(f)0 −mp(f)∞ + 1 = mp(f)0 −m+ 1 ≥ 0

8

requires the existence of Dm+1, . . . , D2m−1 ⊂ (f)0 =k∑

i=1

Di with p ∈ Dm+1 ∩ . . . ∩

D2m−1, after a further permutation of Dm+1, . . . , Dk. Now 2m − 1 ≤ k impliesthat mp(f)∞ = m ≤

[k+12

].

In particular, for k = 2 the inequality 2 ≤ mp(f)∞ ≤[32

]cannot be satisfied.

Proposition 7. If

f(u, v) =3∏

i=1

σ(aiu+ biv + ci + µi)

σ(aiu+ biv + ci)(1.7)

is a special σ-quotient from LA(T, Tsing) then T1 ∩T2 ∩T3 = ∅ and the intersection

numbers T1.T2 = T2.T3 = T3.T1 ∈ N are equal.

Proof. By Lemma 6 there follows mp(f)∞ = 2 for ∀p ∈ (f)sing∞ . In particular,(f)∞ = T1 + T2 + T3 has no triple point and T1 ∩ T2 ∩ T3 = ∅. Further, for anyp ∈ Ti ∩ Tj the condition mp(f) + 1 ≥ 0 requires p ∈ Dl, so that µl ∈ O−d −∆−1

ij ∆ and ∆−1ij ∆jl,∆

−1ij ∆li ∈ O−d, according to Lemma 1 (ii). A cyclic change

of the even permutation i, j, l by j, l, i and l, i, j results in ∆−1jl ∆li,∆

−1jl ∆ij ∈ O−d

and, respectively, ∆−1li ∆ij ,∆

−1li ∆jl ∈ O−d. Consequently, ∆

−1ij ∆jl,∆

−1ij ∆li ∈ O∗

−d,

whereas NQ(

√−d)

Q (∆ij) = NQ(

√−d)

Q (∆jl) = NQ(

√−d)

Q (∆li). Now, by Lemma 1 (i)there follows Ti.Tj = Tj .Tl = Tl.Ti.

Definition 8. The divisor T1 + T2 + T3 with three smooth elliptic irreduciblecomponents is called a triangle if T1 ∩ T2 ∩ T3 = ∅ and T1.T2 = T2.T3 = T3.T1 = 1.

Examples of special σ-quotients with triangular pole divisors are constructedby Holzapfel in [4]. We show that any triangular divisor can be realized as a poledivisor of a special σ-quotient f ∈ LA(T, T

sing) and provide a general formula forsuch f .

Proposition 9. Let Ti = (u + O−d, v + O−d) | a′iu + b′iv + c′i ∈ O−d with 1 ≤i ≤ 3 be the smooth irreducible elliptic components of a triangle T1 + T2 + T3 andai = ∆′

jla′i, bi = ∆′

jlb′i, ci = ∆′

jlc′i. Then a1 + a2 + a3 = 0, b1 + b2 + b3 = 0,

∆−112 ∆ ∈ O−d and for any ν ∈ O−d the function

f(u, v) =

3∏i=1

σ(aiu+ biv + ci −∆−112 ∆+ ν)

σ(aiu+ biv + ci)(1.8)

is a special σ-quotient from LA(T, Tsing) with pole divisor (f)∞ = T1 + T2 + T3.

9

Proof. Let v′i =

(a′ib′i

)for 1 ≤ i ≤ 3. Expanding along the third row, one obtains

0 =

∣∣∣∣∣∣a′1 a′2 a′3b′1 b′2 b′3a′1 a′2 a′3

∣∣∣∣∣∣ = ∆′23a

′1 +∆′

31a′2 +∆′

12a′3 = 0,

0 =

∣∣∣∣∣∣a′1 a′2 a′3b′1 b′2 b′3b′1 b′2 b′3

∣∣∣∣∣∣ = ∆′23b

′1 +∆′

31b′2 +∆′

12b′3 = 0,

and concludes that

v1 + v2 + v3 = ∆′23v

′1 +∆′

31v′2 +∆′

12v′3 = 02×1, ∆12 = ∆23 = ∆31. (1.9)

Now, according to Lemma 1 (iii), T1 ∩ T2 ∩ T3 = ∅ is equivalent to ∆ ∈ ∆12O−d.Then the condition mp(f)0 ≥ mp(f)∞ − 1 for ∀p ∈ (f)sing∞ reduces to Ti ∩ Tj ⊂ Dl

for any even permutation i, j, l of 1, 2, 3. Making use of Lemma 1 (ii), one canchoose µ1 = µ2 = µ3 = ν −∆−1

12 ∆ ∈ O−d. Then (1.9) implies (1.4) from Lemma 4and reveals that (1.8) is a special σ-quotient from LA(T, T

sing).

Definition 10. The special σ-quotients (1.8) from LA(T, Tsing) with triangular

pole divisors (f)∞ = T1 + T2 + T3 are called triangular.

For elliptic curves Ti = (u+O−d, v +O−d) | aiu+ biv + ci ∈ O−d, 1 ≤ i ≤ 2with minimal fundamental groups π1(Ti) = π1(E) = O−d and intersection numberT1.T2 = 1, Lemma 1 (i) implies that

M =

(a2 b2a1 b1

)∈ GL2(O−d).

As a result, there arises an automorphism

φ : A −→ A,

φ(u+O−d, v +O−d) =

[M

(uv

)+

(c2c1

)]twith φ(T1) = E × oE , φ(T2) = oE × E. Making use of σ(αz) = ασ(z) for ∀α ∈O∗

−d, ∀z ∈ C, one observes that any triangular σ-quotient can be reduced by anautomorphism of A to the form

f012(u, v) =σ(u+ a−1

0 c0)σ(v + b−10 c0)σ(a0u+ b0v)

σ(u)σ(v)σ(a0u+ b0v + c0)(1.10)

with a0, b0 ∈ O∗−d, c0 ∈ O−d.

We are going to describe the complete divisor of a triangular σ-quotient.

10

Definition 11. The divisor D =2∑

i=0

Di −2∑

i=0

Ti with smooth elliptic irreducible

components Di, Tj is called a tetrahedron (cf. Figure 1) if:

(i)2∑

i=0

Ti is a triangle;

(ii) Di are parallel to Ti for all 0 ≤ i ≤ 2;(iii) D0 ∩ D1 ∩ D2 = D0 ∩ D1 = D1 ∩ D2 = D2 ∩ D0 = p0 for some point

p0 ∈ A and

(iv)

(2∑

i=0

Di

)∩(

2∑i=0

Ti

)=

(2∑

i=0

Ti

)sing

⊂(

2∑i=0

Di

)smooth

.

T0

T1 T2

D

D

D2

0

1

Figure 1: Tetrahedron

Definition 12. An inscribed (ordered) pair of triangles (cf. Figure 2) is a divisor

D =2∑

i=0

Di −2∑

i=0

Ti, such that:

(i)2∑

i=0

Di and2∑

i=0

Ti are triangles;

(ii) Di are parallel to Ti for all 0 ≤ i ≤ 2 and

(iii)

(2∑

i=0

Di

)∩(

2∑i=0

Ti

)=

(2∑

i=0

Ti

)sing

⊂(

2∑i=0

Di

)smooth

.

An explicit calculation of the singular points of the complete divisor yields thefollowing

Corollary 13. Let (1.10) with a0, b0 ∈ O∗−d, c0 ∈ O−d be a triangular σ-quotient

with complete divisor (f012) =2∑

i=0

Di −2∑

i=0

Ti. Then:

11

‘

D0

T0

T1

T2

D

D2

1

Figure 2: Inscribed (ordered) pair of triangles

(i) c0 +O−d ∈ E2−tor is a 2-torsion point if and only if (f012) is a tetrahedron;(ii) c0 +O−d ∈ E2−tor exactly when (f012) is an inscribed pair of triangles.In either case, the multiplicity mp(f012) = −1 at all p ∈ (f012)∞ ∩ T sing.

In [4] Holzapfel introduces the idea for detecting the linear independence ofco-abelian modular forms by the poles of the corresponding transfers to abelianfunctions. Instead of his strongly descending divisor condition, we use a naturalcomplete decreasing flag on [Γ, 1]. That enables to supply a criterion for somemodular forms to constitute a basis of [Γ, 1] and to show that [Γ, 1] has always abasis of the considered form.

Observe that the subspaces

Vi = j1[Γ, 1]i := ω ∈ j1[Γ, 1] |ω(κ1) = . . . = ω(κi−1) = 0

of V1 = j1[Γ, 1] form a non-increasing flag

j1[Γ, 1] = V1 ⊇ V2 ⊇ . . . ⊇ Vm−1 ⊇ Vm ⊇ . . . ⊇ Vh ⊇ Vh+1 = j1[Γ, 1]cusp.

For any ω, ω′ ∈ Vi one has ω′(κi)ω−ω(κi)ω

′ ∈ Vi+1, so that 0 ≤ dimC (Vi/Vi+1) ≤ 1for all 1 ≤ i ≤ h. We prove that there is a permutation of the cusps κ1, . . . , κh, sothat Vi/Vi+1 ≃ C for 1 ≤ i ≤ m and Vm+1 = Vm+2 = . . . = Vh+1 = j1[Γ, 1]cusp ≃ C.If so, then dimC[Γ, 1] = m+ 1.

Proposition 14. If the pole divisors of fi ∈ LA(T, Tsing) are subject to

Ti ⊂ (fi)∞ ⊆ Ti + Ti+1 + . . .+ Th for all 1 ≤ i ≤ m,

then ωi = τ−11 (fi) ∈ j1[Γ, 1] with 1 ≤ i ≤ m form a basis of a complement of

Vm+1 = j1[Γ, 1]m+1.In particular, if Vm+1 = Vh+1 = j1[Γ, 1]cusp, then j1(ηo), ω1, . . . , ωm is a C-basis

of j1[Γ, 1].

12

Proof. It suffices to show that for arbitrary b1, . . . , bm ∈ C the linear system

m∑i=1

ωi(κj)ti = bj , 1 ≤ j ≤ m (1.11)

has unique solution (t1, . . . , tm). On one hand, that implies the linear independenceof ω1, . . . , ωm over C. On the other hand, for any ω ∈ j1[Γ, 1] there is uniquely

determinedm∑i=1

ciωi with ω0 = ω −m∑i=1

ciωi ∈ j1[Γ, 1]m+1 = Vm+1. In other words,

j1[Γ, 1] = SpanC(ω1, . . . , ωm) ⊕ Vm+1, so that ω1, . . . , ωm is a basis of the comple-ment SpanC(ω1, . . . , ωm) of Vm+1.

Towards the existence of a unique solution of (1.11), note that Ti ⊂ (τ1(ωi))∞ ⊆Ti + Ti+1 + . . . + Th is equivalent to ωi(κi) = 0 and ωi(κ1) = ωi(κ2) = . . . =ωi(κi−1) = 0. Thus, (1.11) is of the form

ω1(κ1) . . . 0 . . . 0

ω1(κi) . . . ωi(κi) . . . 0

ω1(κm) . . . ωi(κm) . . . ωm(κm)

t1...ti...tm

=

b1...bi...bm

,

with non-degenerate, lower-triangular coefficient matrix and has unique solutionfor all b1, . . . , bm ∈ C.

In the case of Vm+1 = Vh+1 = j1[Γ, 1]cusp, note that j1[Γ, 1]cusp = Cj1(ηo) withτ1j1(ηo) = 1 ∈ LA(T, T

sing), so that j1(ηo), ω1, . . . , ωm is a C-basis of j1[Γ, 1].

The next proposition establishes that j1[Γ, 1] has always a C-basis of the con-sidered form.

Proposition 15. Let Γ ⊂ SU2,1 be a freely acting, co-abelian Picard modular groupand dimC[Γ, 1] = m + 1. Then there is a permutation κ1, . . . , κm, κm+1, . . . , κh ofthe Γ-cusps, such that

V1/V2 ≃ V2/V3 ≃ . . . ≃ Vm/Vm+1 ≃ C, Vm+1 = Vm+2 = . . . = Vh+1 = j1[Γ, 1]cusp.

Any ωi ∈ Vi \ Vi+1 transfers to τ1(ωi) ∈ LA(T, Tsing) with

Ti ⊂ (τ1(ωi))∞ ⊆ Ti + Ti+1 + . . .+ Th for 1 ≤ i ≤ m

and j1(ηo), ω1, . . . , ωm is a C-basis of V1 = j1[Γ, 1].In particular, if Th−1.Th = 1 then Vh−1 = j1[Γ, 1]cusp and dim[Γ, 1] ≤ h− 1.

Proof. If V1 = Vh+1, then there is nothing to be proved. From now on, we assumethat dimV1/Vh+1 = m ∈ N. By an induction on 1 ≤ i ≤ m, we establish theexistence of ωj ∈ Vj \ Vj+1 for all 1 ≤ j ≤ i. First of all, for any ω1 ∈ V1 \ Vh+1

13

there exists a cusp κ1 with ω1(κ1) = 0. Then for an arbitrary permutation ofthe remaining cusps, one has ω1 ∈ V1 \ V2. If we have chosen ωj ∈ Vj \ Vj+1 for1 ≤ j ≤ i − 1 and Vi ! Vh+1, then for an arbitrary ωi ∈ Vi \ Vh+1 there existsa permutation of κi, κi+1, . . . , κh, such that ωi(κi) = 0. Clearly, ωi ∈ Vi \ Vi+1

and we have obtained a basis j1(ηo), ω1, . . . , ωm of V1 = j1[Γ, 1]. The conditionsωi ∈ Vi \ Vi+1 amount to Ti ⊂ (τ1(ωi))∞ and Tj " (τ1(ωi))∞ for all 1 ≤ j ≤ i− 1.

If Th−1.Th = 1 then up to an automorphism of A, one can assume that Th−1 =E × oE and Th = oE × E. We claim that LA((E × oE) + (oE × E)) = C, sothat dimC[Γ, 1] = m + 1 ≤ h − 1. Indeed, for an arbitrary Q ∈ E \ oE therestriction f |E×Q is an elliptic function of order 1. Therefore f |E×Q ≡ C(Q) ∈ Cis constant. Similarly, f |P×E ≡ C ′(P ) ∈ C for any P ∈ E \ oE . As a result,C ′(P ) = f(P,Q) = C(Q) for all Q ∈ E and f |A is constant.

Proposition 16. (Holzapfel [5]) Let us fix the half-periods ω1 = 12 , ω2 = i

2 ,ω3 = ω1 + ω2 of the lattice π1(E) = O−1 = Z + iZ, the 2-torsion points Q0 :=0(modZ+ iZ) ∈ E, Qj := ωj(modZ+ iZ) ∈ E for 1 ≤ j ≤ 3 and Qij := (Qi, Qj) ∈A. Consider the elliptic curves

Tk = (u+ π1(E), v + π1(E)) |u− ikv ∈ π1(E) for 1 ≤ k ≤ 4,

T4+k = (u+ π1(E), v + π1(E)) |u− ωk ∈ π1(E) for 1 ≤ k ≤ 2,

T6+k = (u+ π1(E), v + π1(E)) | v − ωk ∈ π1(E) for 1 ≤ k ≤ 2.

Then the blow-up of A at the singular points

S1 = Q00, S2 = Q33, S3 = Q11, S4 = Q12, S5 = Q21, S6 = Q22

of T(6,8)√−1

=8∑

k=1

Tk is the toroidal compactification (B/Γ1)′of a ball quotient B/Γ1

by a freely acting Picard modular group Γ1 over the Gaussian integers Z[i].

The self-intersection matrix M(6, 8) ∈ Z6×8 of T(6,8)√−1

is defined to have entries

M(6, 8)ij = 1 for Si ∈ Tj and M(6, 8)ij = 0 for Si ∈ Tj . Straightforwardly,

M(6, 8) =

1 1 1 1 0 0 0 01 1 1 1 0 0 0 00 1 0 1 1 0 1 01 0 1 0 1 0 0 11 0 1 0 0 1 1 00 1 0 1 0 1 0 1

.

According to Q00, Q33 ∈ Tk or ∀1 ≤ k ≤ 4, there are no triangles Ti + Tj +

Tk ⊂ T(6,8)√−1

with 1 ≤ i < j ≤ 4, 1 ≤ i < j < k ≤ 8. Bearing in mind that(T

(6,8)√−1

)sing∩(

8∑k=5

Tk

)=

2∑i=1

2∑j=1

Qij , one makes use of Figure ?? and recognizes

14

Q Q

Q

T

T

T

T

T

T

T

T

22

11 12

1

3

4

2

7

6

8

5

Q11

Figure 3: The incidence relations of T(6,8)√−1

and2∑

i=1

2∑j=1

Qij ⊂(T

(6,8)√−1

)sing.

the triangles T2k−1 + T4+m + T6+m, T2k + T4+m + T9−m with 1 ≤ k,m ≤ 2.An immediate application of Proposition 9 with ν = 2ωm and, respectively, ν =ω3 + ωm + (−1)k+1ω3−m, yields the following

Corollary 17. The space LA

(T

(6,8)√−1

,(T

(6,8)√−1

)sing)contains the binary parallel

f56(u, v) =σ(u− ω1 − µ1)σ(u− ω2 + µ1)

σ(u− ω1)σ(u− ω2),

f78(u, v) =σ(v − ω1 − µ2)σ(v − ω2 + µ2)

σ(v − ω1)σ(v − ω2)

and the triangular σ-quotients

f2k−1,4+m,6+m(u, v) =

σ(u+ (−1)kiv + ω3)σ(−u+ ωm + ω3)σ((−1)k+1iv + (−1)kiωm + ω3)

σ(u+ (−1)kiv)σ(−u+ ωm)σ((−1)k+1iv + (−1)kiωm)

f2k,4+m,9−m(u, v) =

σ(u+ (−1)k+1v + ω3)σ(−u+ ωm + ω3)σ((−1)kv + (−1)k+1ω3−m + ω3)

σ(u+ (−1)k+1v)σ(−u+ ωm)σ((−1)kv + (−1)k+1ω3−m)

with arbitrary 1 ≤ k,m ≤ 2.

15

Proposition 14 provides the following

Corollary 18. If fpq and fijk are the binary parallel and triangular σ-quotients

from the space LA

(T

(6,8)√−1

,(T

(6,8)√−1

)sing)and ωpq = τ−1

1 (fpq), ωijk = τ−11 (fijk),

thenω157, ω258, ω368, ω467, ω56, ω78, j1(ηo)

is a C-basis of j1[Γ1, 1].In particular, dimC[Γ, 1] = 7.

2. SUFFICIENT CONDITIONS FOR THE NORMAL GENERATION OF ASPACE OF LOGARITHMIC CANONICAL SECTIONS

Definition 19. A holomorphic line bundle E on an algebraic variety X is suffi-ciently ample if the holomorphic sections of a sufficiently large tensor power E⊗m

provide a projective embedding of X.

Definition 20. A holomorphic line bundle E over an algebraic variety X is globallygenerated if the global holomorphic sections of E determine a regular projectivemorphism.

A subspace V ⊆ H0(X, E) is globally generated if some (and therefore any)basis of V provides a regular projective morphism X → P(V ).

Definition 21. A holomorphic line bundle E over an algebraic manifold X isnormally generated if E is globally generated and H0(X, E) defines a projectiveimmersion of X with normal image.

A subspace V ⊆ H0(X, E) is normally generated if it is globally generated andthe morphism X → P(V ) is a projective immersion with normal image.

The normal generation of a sufficiently ample line bundle is a classical topic un-der study. Various works provide normally generated and non-normally generatedline bundles over curves and abelian varieties. According to [2], if E is a sufficientlyample line bundle on an abelian variety of dimension n, then E⊗(n−1) is normallygenerated. In particular, any sufficiently ample line bundle on an abelian surfaceis normally generated.

Our aim is to provide sufficient conditions for the normal generation of a sub-space V ⊆ H0(A′,Ω2

A′(T ′)) over the Baily-Borel compactification A. That cannotbe derived from the normal generation of a subspace W ⊆ H0(A, E) of holomorphicsections of a line bundle E → A. Namely, the morphism, associated with the pull-back ξ∗W is not immersive on the exceptional divisor L = ξ−1(T sing) of ξ : A′ → A.Moreover, ξ∗W is non-constant on T ′

i , as far as W is non-constant on Ti. Thus, the

restriction of ξ∗W on A′ \ T ′ = B/Γ = B/Γ \ (∂ΓB/Γ) does not extend to A = B/Γand cannot yield the normal generation of V ⊆ H0(A′,Ω2

A′(T ′)) over A.

16

Lemma 22. Any regular immersion f : X → Y of algebraic varieties is birationalonto its image f(X).

Proof. The surjective regular morphism f : X → f(X) induces an injective C-algebra homomorphism f∗ : C[f(X)] → C[X] of the rings of regular functions. Itextends to an embedding f∗ : C(f(X)) → C(X) of the corresponding meromorphicfunction fields, which are the fraction fields of the rings of the regular functions. Itsuffices to prove that f∗C(f(X)) = C(X), in order to conclude that f : X → f(X)is birational. More precisely, f : X → f(X) is a local biholomorphism at anypoint x ∈ X. Consequently, f induces an isomorphism f∗ : Of(x),f(X) → Ox,X

of the local rings. For any φ ∈ C(X) there exists x ∈ X at which φ is regular.Therefore φ ∈ Ox,X = f∗Of(x),f(X) ⊆ f∗C(f(X)) and f∗ : C(f(X)) → C(X) is anisomorphism of the meromorphic function fields.

Corollary 23. The following are equivalent for an irreducible normal projectivevariety X:

(i) f : X → Y is a regular immersion with normal image Y ;(ii) f : X → Y is a finite, regular, generically injective morphism onto Y ;(iii) f : X → Y is biregular.

Proof. (i) =⇒ (ii) By Lemma ??, any surjective regular immersion f : X → Yis birational. In particular, f is generically injective. The fibers of an immersionf : X → Y are discrete, as far as the differentials (df)x : T 1,0

x X → T 1,0f(x)Y vanish

on the tangent spaces T 1,0x f−1(f(x)) of the fibers f−1(f(x)). The compactness of

X implies the compactness of all the fibers f−1(y), y ∈ Y . As a result, f−1(y) arediscrete and compact, whereas finite.

(ii) =⇒ (iii) If f : X → Y is a regular morphism of degree d ∈ N, then thegeneric fiber of f consists of d points, while the exceptional ones are constituted by≤ d points. In particular, for d = 1, any regular, generically injective morphism isbijective onto its image.

(iii) =⇒ (i) is obvious.

Our specific considerations will be based on the following immediate conse-quence of Corollary ??

Corollary 24. Let X be an irreducible normal projective variety, E → X be eholomorphic line bundle over X and V ⊆ H0(X, E) be a space of global holomorphicsections of E. Then V is normally generated if and only if it determines a finite,regular, generically injective morphism f : X → P(V ).

Lemma 25. A subspace V ⊆ H0(A′,Ω2A′(T ′)), containing the cuspidal form j1(ηo),

is globally generated over A if and only if it satisfies simultaneously the followingtwo geometric conditions:

(i) for any irreducible component Ti of T there is ωi ∈ V with (τ1(ωi))∞ ⊃ Ti;(ii) for any p ∈ T sing there exists ωp ∈ V with mp(τ1(ωp)) = −1.

17

Proof. The space V is globally generated over A exactly when for any point q ∈ A

there is vq ∈ V with vq(q) = 0. If q ∈(B/Γ

)\(L ∪

h∑i=1

κi

)then j1(ηo)(q) = 0. A

modular form ωi ∈ V does not vanish on the cusp κi if and only if Ti ⊂ (τ1(ωi))∞.A modular form ωp ∈ V takes non-zero values on the rational (−1)-curve ξ−1(p)exactly when the multiplicity mp(τ1(ωp)) = −1.

From now on, we say briefly that a modular form ω ∈ H0(A′,Ω2A′(T ′)) is bi-

nary parallel or triangular if its transfer τ1(ω) ∈ LA(T, Tsing) is binary parallel or,

respectively, triangular.

Proposition 26. Let us suppose that the subspace V ⊆ H0(A′,Ω2A′(T ′)) contains

the cuspidal form j1(ηo), two binary parallel forms ω13, ω24, a triangular ω012 withT0 ∩ T3 ∩ T4 = ∅ and satisfies the following three conditions:

(i) for any i ∈ 0, 1, . . . , 4 there exists ωi ∈ V with (τ1(ωi))∞ ⊃ Ti,

(ii) for any p ∈ T sing \

(4∑

j=0

Tj

)there exists ωp ∈ V with mp(τ1(ωp)) = −1 and

(iii) for any 1 ≤ i < j ≤ h there is ωij ∈ V , such that (τ1(ωij))∞ containsexactly one of Ti or Tj.

Then V is normally generated.

Proof. In the presence of Corollary ??, it suffices to establish that the projectivemorphism f : A → P(V ), associated with V is regular, finite and generically in-jective. Assumption (i) from the present proposition and (τ1(ωij))∞ = Ti + Tj ,(τ1(ω012))∞ = T0 + T1 + T2 imply assumption (i) from Lemma ??. Further, no-one p ∈ T sing ∩ (T1 + T3) belongs to (τ1(ω13))0 = D1 + D3, as far as T1, T3, D1

and D3 are mutually parallel and distinct. Therefore mp(τ1(ω13)) = −1. Similarly,mp(τ1(ω24)) = −1 for p ∈ T sing∩(T2+T4). By Corollary 13, mp(τ1(ω012)) = −1 for

all p ∈ T sing ∩(

2∑i=0

Ti

). Combining with assumption (ii) from the present proposi-

tion, one obtains (ii) from Lemma ??. That allows to conclude that f : A → P(V )is regular.

The assumption (iii) guarantees that f : A → f(A) ⊂ P(V ) is finite. First ofall, on A \ [L+ (∂ΓB/Γ)] = (B/Γ) \ L = A \ T , the morphism(

ω13

j1(ηo)= f13 ξ = f13,

ω24

j1(ηo)= f24 ξ = f24

): (B/Γ) \ L = A \ T −→ C2

is of degree 4. More precisely, if

f13(u, v) =σ(u− µ1)σ(u− c3 + µ1)

σ(u)σ(u− c3), f24(u, v) =

σ(v − µ2)σ(v − c4 + µ2)

σ(v)σ(v − c4), (2.1)

then for any x, y ∈ P1 the fiber is

(f13, f24)−1 = (Pi(x), Qj(y)) | 1 ≤ i, j ≤ 2 with

18

P1(x) + P2(x) = c3, Q1(y) +Q2(y) = c4.

The condition (iii) provides the injectiveness of f : ∂ΓB/Γ → f (∂ΓB/Γ), whichsuffices for f : L → f(L) to be discrete and, therefore, finite. Otherwise, f contractssome irreducible component ξ−1(p), p ∈ T sing of L. If p ∈ Ti ∩ Tj then κi, κj ∈ξ−1(p), whereas f(κi) = f(κj). Thus, f : L ∪ (∂ΓB/Γ) → f (L ∪ (∂ΓB/Γ)) and,

therefore, f : A → f(A) is a finite regular morphism.The generic injectiveness of the projective morphism f : A → f(A) follows from

the generic injectiveness of the affine morphism

F =

(ω13

j1(ηo)= f13,

ω24

j1(ηo)= f24,

ω012

j1(ηo)= f012

): (B/Γ) \ L = A \ T −→ C3.

This, in turn, is equivalent to the generic injectiveness of the rational surjectivemorphism

F = (f13, f24, f012) : A −→ P1 × P1 × P1.

Let us consider also the rational surjection F1 = (f13, f24) : A → P1 × P1 and itsfactorization

A P1 × P1 × P1

P1 × P1

?F1

-F

+

pr12

through F and the projection pr12 : P1×P1×P1 → P1×P1 onto the first two factors.The irreducible components T1 and T2 of the triangle T0+T1+T2 have intersectionnumber T1.T2 = 1. That allows to assume that T1 = oE × E, T2 = E × oE and(1.10).

Suppose that F : A → P1×P1×P1 is not generically injective. By F1 = pr12 Fand degF1 = 4, the generic fiber of F on F−1

1 (x, y) consists of 2 or 4 points. Ineither case, for any (x, y) ∈ P1 × P1 there holds at least one of the following pairsof relations:

Case (i): f012(P1(x), Q2(y)) = f012(P2(x), Q1(y)),f012(P1(x), Q1(y)) = f012(P2(x), Q2(y));

Case (ii): f012(P1(x), Q2(y)) = f012(P2(x), Q2(y)),f012(P1(x), Q1(y)) = f012(P2(x), Q1(y));

Case (iii): f012(P1(x), Q2(y)) = f012(P1(x), Q1(y)),f012(P2(x), Q2(y)) = f012(P2(x), Q1(y)).

We claim that the relations from at least one case are satisfied identically onP1×P1. Otherwise, the locus of either case is a proper analytic subvariety of P1×P1

and their union is also a proper analytic subvariety of P1 × P1. The contradictionimplies that for any (x, y) ∈ P1×P1 there holds identically at least one of the Cases(i), (ii) or (iii). Note that (ii) and (iii) are equivalent under the transposition of thefactors of P1 × P1 and, respectively, of A = E × E.

19

Without loss of generality, one can suppose that P1(∞) = oE and P2(∞) = c3.In Case (i), after an eventual relabeling of Q1(y), Q2(y), one has Q1(∞) = oE ,Q2(∞) = c4. Then

∞ = f012(oE , oE) = f012(P1(∞), Q1(∞)) = f012(P2(∞), Q2(∞)) = f012(c3, c4).

However, c3 = oE , c4 = oE and T3 ∩ T4 = (c3, c4) " T0 reveal that f012(c3, c4) =∞, so that Case (i) does not hold identically on A. Similarly, in Case (ii), therefollows

∞ = f012(oE , c4) = f012(P1(∞), Q2(∞)) = f012(P2(∞), Q2(∞)) = f012(c3, c4).

The contradiction implies that F : A → P1 × P1 × P1 is generically injective.

Here is another sufficient condition for a subspace V ⊆ H0(A′,Ω2A′(T ′)) to be

normally generated.

Proposition 27. Let V be a subspace of H0(A′,Ω2A′(T ′)), containing the cuspidal

form j1(ηo), a binary parallel ω13, triangular ω012, ω234 with T0 ∩ T1 ∩ T4 = ∅ andsatisfying the following three conditions:

(i) for any i ∈ 0, 1, . . . 4 there exists ωi ∈ V with (τ1(ωi))∞ ⊃ Ti;

(ii) for any p ∈ T sing \

(4∑

j=0

Tj

)there exists ωp ∈ V with mp(τ1(ωp)) = −1;

(iii) for any 1 ≤ i < j ≤ h there is ωij ∈ V , such that (τ1(ωij))∞ containsexactly one of Ti or Tj.

Then V is normally generated.

Proof. As in Proposition ??, first we establish the regularity of the projective mor-phism f : A → f(A).

Further, f : A → f(A) is finite, as far as the fibers of its restriction on (B/Γ) \L = A \ T are contained in the fibers of(

ω13

j1(ηo)= f13,

ω012

j1(ηo)= f012

): A \ T −→ C2.

Let f012(u, v) be of the form (1.10) and f13 be as in (??). Then for any x, y ∈ P1

the fiber

(f13, f012)−1(x, y) = (Pi(x), Qij(x, y)) | 1 ≤ i, j ≤ 2, with

P1(x) + P2(x) = c3, Qi1(x, y) +Qi2(x, y) = −a0b−10 Pi(x)− b−1

0 c0

consists of at most four points. The reason is that for any fixed Pi(x) ∈ E theelliptic function f012(Pi(x), ) is of order 2. Thus, (f13, f012) : A \ T → C2 is finite.The assumption (iii) implies that f : L ∪ (∂ΓB/Γ) → f (L ∪ (∂ΓB/Γ)) is finite, sothat f : A → f(A) is a finite regular morphism.

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We derive the generic injectiveness of f : A → f(A) from the generic injective-ness of the affine morphism

F =

(ω13

j1(ηo)= f13,

ω012

j1(ηo)= f012,

ω234

j1(ηo)= f234

): (B/Γ) \ L = A \ T −→ C3.

To this end, let us factor the rational surjection F1 = (f13, f012) : A → P1 × P1

through the rational surjection F = (f13, f012, f234) : A → P1 × P1 × P1 and theprojection pr12 : P1 × P1 × P1 → P1 × P1, along the commutative diagram

A P1 × P1 × P1

P1 × P1

?F1

-F

+

pr12.

If F is not generically injective, then at least one of the following three cases holdsidentically on P1 × P1:

Case (i): f234(P1(x), Q12(x, y)) = f234(P2(x), Q21(x, y)),f234(P1(x), Q11(x, y)) = f234(P2(x), Q22(x, y));

Case (ii): f234(P1(x), Q12(x, y)) = f234(P2(x), Q22(x, y)),f234(P1(x), Q11(x, y)) = f234(P2(x), Q21(x, y));

Case (iii): f234(P1(x), Q12(x, y)) = f234(P1(x), Q11(x, y)),f234(P2(x), Q22(x, y)) = f234(P2(x), Q21(x, y)).

In either case, denote by P1(∞) = oE and P2(∞) = c3 the poles of the ellipticfunction f13 and note that T1 = P1(∞) × E, T3 = P2(∞) × E. Further, letQi1(∞,∞) = oE , so that T2 = E × Q11(∞,∞) = E × Q21(∞,∞). Finally, letQi2(∞,∞) = −a0b

−10 Pi(∞)− b−1

0 c0, in order to have

q10 = T1 ∩ T0 = (P1(∞), Q12(∞,∞)),

q30 = T3 ∩ T0 = (P2(∞), Q22(∞,∞)).

Denote alsoq12 = T1 ∩ T2 = (P1(∞), Q11(∞,∞)),

q32 = T3 ∩ T2 = (P2(∞), Q21(∞,∞)).

Bearing in mind that (f234)∞ = T2 + T3 + T4, note that f234(qij) = ∞ wheneveri, j∩ 2, 3, 4 = ∅. In the Case (i) one has f234(q10) = f234(q32) = ∞. If q10 ∈ T2

then q10 ∈ T0 ∩ T1 ∩ T2, contrary to the assumption that T0 + T1 + T2 is a triangle.On the other hand, T3 ∩ T1 = ∅ guarantees that q10 ∈ T3. Therefore q10 ∈ T4 andq10 ∈ T0 ∩ T1 ∩ T4 = ∅. The contradiction rejects the Case (i). If the first relationof Case (ii) is identical on P1 × P1 then f234(q10) = f234(q30) = ∞. As in theCase (i), that leads to an absurd. Finally, f234(q10) = f234(q12) = ∞ contradictsthe hypotheses and establishes that F = (f13, f012, f234) : A → P1 × P1 × P1 isgenerically injective.

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An immediate application of Proposition ?? to the example from Proposition?? yields the following

Corollary 28. In terms of Proposition ??, the subspace

V1 = SpanC(j1(ηo), ω56, ω157, ω267, ω368, ω458) ⊂ H0(A′1,Ω

2A′

1(T ′))

is normally generated, i.e., determines a regular projective immersion

f : B/Γ1 → P(V1) = P5

with normal image.

If one applies Proposition ?? to the cuspidal form j1(ηo), the binary parallelω56, ω78 and triangular ω157, then one needs to adjoin the triangular ω2,4+k,9−k,ω3,4+l,6+l, ω4,4+m,9−m for some k, l,m ∈ 1, 2. The span of these modular forms is7-dimensional and depletes the entire [Γ1, 1]. It is clear that the normal generationof V1 implies the normal generation of H0(A′

1,Ω2A′

1(T ′)) = j1[Γ1, 1].

3. REFERENCES

1. Borevich, Z.I., Shafarevich I.R., Number Theory, Nauka, 1985.

2. Ewald G., U. Wessels, On the ampleness of invertible sheaves in completeprojective toric varieties, Results in Mathematics, 19 (1991), 275-278.

3. Hemperly J., The parabolic contribution to the number of independent auto-morphic forms on a certain bounded domain, Amer. Jour. Math 94 (1972),1078-1100

4. Holzapfel R.-P., Jacobi theta embedding of a hyperbolic 4-space with cusps, In: Third International Conference in Geometry, Integrability and Quantization,Varna, June 14-23, 2001, pp. 11-63.

5. Holzapfel R.-P., Complex hyperbolic surfaces of abelian type, Serdica Math.Jour. 30 (2004), 207-238.

6. Lang S., Elliptic Functions, Addison-Wesley Publ. Comp., 1973.

7. Privalov I.I., Introduction In The Theory Of Functions Of One ComplexVariable, 1940.

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